Robust Optimization - eaton.math.rpi.edueaton.math.rpi.edu/.../16C_robustbeameripad.pdf · Mitchell...

Robust Optimization

John E. Mitchell

Department of Mathematical SciencesRPI, Troy, NY 12180 USA

March 2020

Mitchell Robust Optimization 1 / 18

Robust LPs

Outline

1 Robust LPs

2 Box-constrained entries

3 Ellipsoid-constrained data


Robust LPs

Robust LPs

We want to solve an LP with constraints Ax � b,but the data is uncertain.

minx cT xsubject to Ax � b 8 A 2 S

x � 0

for some uncertainty set S, typically a box or ellipsoid.

We have uncountably many scenarios: all A in an ellipsoid or a box.

We set up a subproblem: for a given x , what is the worst possible A?

With these structured uncertainty sets, can find the value Ax of theworst scenario explicitly, as a function of x .


Box-constrained entries

Outline

1 Robust LPs






Each entry falls in the range [aij � aij , aij + aij ], with aij � 0.

This can be modelled mathematically, where we replace each elementof A by its lower bound.

minx cT xsubject to

Pnj=1 (aij � aij)xj � bi 8 i

x � 0


ai;ia¥aii→ai;tai;

o

¥2,aiixi.§,aijxi3 b i t i-


Example

Nominal constraint: 2x1 + 3x2 � 6, a = [1, 2].

x1

x2

0

2

6

3 6


24.1¥.. -Lathan 3 6Enaipi


Example

Nominal constraint: 2x1 + 3x2 � 6, a = [1, 2].

x1

x2

0

2

6

3 6

Robust constraint:2x1 + 3x2 � x1 � 2x2 = x1 + x2 � 6


-

[ai;x ; Edits


What if only a proportion of the entries can differ?

A more complicated model is when only a proportion of the data differsfrom an expected value [3].

At first glance, this becomes a binary integer program.

By careful modeling, can still formulate the problem asa linear program with a polynomial number of constraints.


:


The Bertsimas-Sim model

For simplicity, we assume the variables are nonnegative. The modelcan be extended to the general case where l x u.

We allow at most �i entries in row i of the constraint matrix to differfrom their nominal values aij .

Si is the set of entries in row i that differ from their nominal values.

So |Si | �i for each row i .

Model:

minx cT xsubject to

Pnj=1 aijxj �

Pj2Si

aij xj � bi 8 |Si | �i , 8 i

x � 0


O

-

Earlier: JE,a i ; x ;-§iaiixi?b i


Rewrite the constraint using a max function

Our robust model is

minx cT xsubject to

Pnj=1 aijxj �

Pj2Si

aij xj � bi 8 |Si | �i , 8 i

x � 0

which is linear, but has an exponential number of constraints.

Can write the problem equivalently as a nonlinear program:

minx cT xsubject to

Pnj=1 aijxj � max|Si |�i

nPj2Si

aij xj

o� bi 8 i

x � 0


- o f -

¥ ¥


Solving the inner max problem parametrically

Our nonlinear formulation:

minx cT xsubject to

Pnj=1 aijxj � max|Si |�i

nPj2Si

aij xj

o� bi 8 i

x � 0

Given x , the max value in the constraint for row i can be found bysolving a linear program:

�i(x) := maxzPn

j=1 aij xj zjsubject to

Pnj=1 zj �i

0 zj 1 8 j

Note: zj = 1 () j 2 Si


0


Example

Nominal constraint: 2x1 + 3x2 � 6, a = [1, 2], � = 1.

x1

x2

0

2

6

3 6

�(x) := maxz x1 z1 + 2x2 z2subject to z1 + z2 1

0 zj 1 j = 1, 2


pin: :#' {iii.' I I . 2xitxi-sicxlz.coaerial 2x.tk.

-;Esiaiixi36HSik1.Lx,-13*-43624+3×-2×16


Example


x1

x2

0

2

6

3 6

(3, 1)

�(x) := maxz x1 z1 + 2x2 z2subject to z1 + z2 1

0 zj 1 j = 1, 2

x = (3, 1) : �(x) = 3, 2x1 +3x2 ��(x) = 6+3�3 = 6 � 6X feasible


•

P (x)= m a x 32,- 122 ,s t . Z ,t r , E l

° E-jet,;=1,2)2¥44

-


Example


x1

x2

0

2

6

3 6

(1, 3)�(x) := maxz x1 z1 + 2x2 z2

subject to z1 + z2 10 zj 1 j = 1, 2

x = (3, 1) : �(x) = 3, 2x1 +3x2 ��(x) = 6+3�3 = 6 � 6X feasible

x = (1, 3) : �(x) = 6, 2x1+3x2��(x) = 2+9�6 = 5<6 X infeasible


ACH= I I ? !762L}sola:z - c o ,l )

B (x)= G

-


Example


x1

x2

0

2

6

3 6

Robust constraints:2x1 + 3x2 � x1 = x1 + 3x2 � 62x1 + 3x2 � 2x2 = 2x1 + x2 � 6

x = (3, 1) : �(x) = 3, 2x1 +3x2 ��(x) = 6+3�3 = 6 � 6X feasible

x = (1, 3) : �(x) = 6, 2x1+3x2��(x) = 2+9�6 = 5<6 X infeasible


Lx,tha-j.Esidiixi3 6

Si:{Rc

si:{2}


Use LP duality to remove the nonlinearity

From LP duality,

�i(x) := maxzPn

j=1 aij xj zjsubject to

Pnj=1 zj �i

0 zj 1 8 j

= minyi ,wi. �i yi +P

j wijsubject to yi + wij � aij xj 8 j

yi � 0, wij � 0 8 j

Here, yi is a scalar variable.

The nonlinearity in the objective disappears with duality.Now the RHS is parametrized by x .


§,aiixi-Bilxbbi V i( l - d a dvyq.int[ d u e l variablewij.it,..-in


Formulating the robust problem as an LPThe robust nonlinear program is

minx cT xsubject to

Pnj=1 aijxj � �i(x) � bi 8 i

x � 0

Using the duality representation for �i(x), this is equivalent to thelinear program

minx ,y ,w cT xsubject to

Pnj=1 aijxj � �i yi �

Pj wij � bi 8 i

aij xj � yi � wij 0 8 i , 8 jx � 0, y � 0, w � 0

Have O(mn) variables and constraints if A is m ⇥ n.

�(x) is a protection function for the constraints, protecting againstuncertainty.


p i(x) :m i n Figitfwijs e .yituij?hi,-xiY i

- f i3 0 ,Wii?O

l d o .-


Example


One constraint, so one component y1. Two variables, so get w11, w12.

minx ,y ,w cT xsubject to 2x1 + 3x2 � y1 � w11 � w12 � 6

x1 � y1 � w11 02x2 � y1 � w12 0x � 0, y � 0, w � 0

x = (3, 1) : feasible with y1 = 2,w11 = 1,w12 = 0.

x = (1, 3) : infeasible for any y1,w11,w12.


t.EE#. -

←mine.fix-iigi-fwipt.in. }hip;-Ji-oiif.fi,. . . -

Need: 2+9-y,- w "-wn76 5 - w, ,7 66 - y. - n o , ,g o }

a d d :

- 6 t o , t o , , 3 0impossible.

Ellipsoid-constrained data

Outline

1 Robust LPs





Second order cone formulationAnother popular model is to use ellipsoids to constrain possiblechoices of aij [2, 1]. With just one constraint:

minx cT xsubject to (a + a)T x � b 8 a satisfying aT M�1a 1

x � 0

For any given x , we have a subproblem to determine if the constraintholds:

mina

{xT a : aT M�1a 1}.

The solution is a = �1pxT Mx

Mx , so the constraint requires

aT x + mina

{xT a : aT M�1a 1} = aT x �p

xT Mx � b.

This is a second order cone constraint.Mitchell Robust Optimization 16 / 18

atxtatxzb.VE

E t .H i spositivedefinite.

-

O- 0

min I k c a , sulu. using KUTcondition..s i t . I ' M -'a 'E l

I +21M-' 2 = 0

0¥, ⇒ i t .# " I

↳ - IAlso,constraint i s active,

s o I'm-'2=1.

So:fj.INT?yfx=l,so2x=Viir#Soa=-¥r# " "

¥⇒=-TIMEOptimal value:

- I T M I


Notes

Can extend to multiple constraints.Uncertainty can be drawn from a lower dimensional space.Here, each ai , ai , yi is a vector:

minx cT xs.t. (ai + ai)

T x � bi 8 ai = Hiyi with yTi M�1

i yi 1, 8 i

x � 0

Equivalent to the second order cone program

minx cT xs.t. aT

i x �q

xT HiMiHTi x � bi 8 i

x � 0



Notes

Can extend to multiple constraints.Uncertainty can be drawn from a lower dimensional space.Here, each ai , ai , yi is a vector:

minx cT xs.t. (ai + ai)

T x � bi 8 ai = Hiyi with yTi M�1

i yi 1, 8 i

x � 0

Equivalent to the second order cone program

minx cT xs.t. aT

i x �q

xT HiMiHTi x � bi 8 i

x � 0


I = II,I ' I I I

A-' IE T


A. Ben-Tal, L. El Ghaoui, and A. Nemirovski.Robust Optimization.Princeton University Press, Princeton, NJ, 2009.

A. Ben-Tal and A. Nemirovski.Robust solutions of uncertain linear programs.Operations Research Letters, 25(1):1–13, 1999.

D. Bertsimas and M. Sim.The price of robustness.Operations Research, 52(1):35–53, 2004.


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